The general principles of operation for a plasma centrifuge are well known and well understood. In short, a plasma centrifuge generates forces on charged particles which will cause the particles to separate from each other according to their mass. More specifically, a plasma centrifuge relies on the effect that crossed electric and magnetic fields have on charged particles. As is known, crossed electric and magnetic fields will cause charged particles in a plasma to move through the centrifuge on respective helical paths around a centrally oriented longitudinal axis. As the charged particles transit the centrifuge under the influence of these crossed electric and magnetic fields they are, of course, subject to various forces. Specifically, in the radial direction, i.e. a direction perpendicular to the axis of particle rotation in the centrifuge, these forces are: 1) a centrifugal force, F.sub.c, which is caused by the motion of the particle; 2) an electric force, F.sub.E, which is exerted on the particle by the electric field, E.sub.r ; and 3) a magnetic force, F.sub.B, which is exerted on the particle by the magnetic field, B.sub.z. Mathematically, each of these forces are respectively expressed as:
F.sub.c =Mr.omega..sup.2 ; PA1 F.sub.E =eE.sub.r ; and PA1 F.sub.B =er.omega.B.sub.z. PA1 M is the mass of the particle; PA1 r is the distance of the particle from its axis of rotation; PA1 .omega. is the angular frequency of the particle; PA1 e is the electric charge of the particle; PA1 E is the electric field strength; and PA1 B.sub.z is the magnetic flux density of the field. PA1 .SIGMA.F.sub.r =0 (positive direction radially outward); PA1 F.sub.c -F.sub.E -F.sub.B =0; EQU Mr.omega..sup.2 -eE.sub.r -er.omega.B.sub.z =0. (Eq. 1) PA1 M.sub.C =e(Ba).sup.2 /(8V.sub.ctr) where we used: EQU .alpha.=(V.sub.ctr -.PHI.)/.PSI.=2V.sub.ctr /(a.sup.2 B.sub.z) (Eq. 2) PA1 .SIGMA.F.sub.r =0 (positive direction radially outward) PA1 F.sub.c +F.sub.E +F.sub.B =0 EQU Mr.omega..sup.2 +eEr-er.omega.B.sub.z =0 (Eq. 3)
Where:
In a plasma centrifuge, it is universally accepted that the electric field will be directed radially inward. Stated differently, there is an increase in positive voltage with increased distance from the axis of rotation in the centrifuge. Under these conditions, the electric force F.sub.E will oppose the centrifugal force F.sub.C acting on the particle, and depending on the direction of rotation, the magnetic force either opposes or aids the outward centrifugal force. Accordingly, an equilibrium condition in a radial direction of the centrifuge can be expressed as:
It is noted that Eq. 1 has two real solutions, one positive and one native, namely: ##EQU1##
where .omega.=eB.sub.z /M.
For a plasma centrifuge, the intent is to seek an equilibrium to create conditions in the centrifuge which allow the centrifugal forces, F.sub.c, to separate the particles from each other according to their mass. This happens because the centrifugal forces differ from particle to particle, according to the mass (M) of the particular particle. Thus, particles of heavier mass experience greater F.sub.c and move more toward the outside edge of the centrifuge than do the lighter mass particles which experience smaller centrifugal forces. The result is a distribution of lighter to heavier particles in a direction outward from the mutual axis of rotation. As is well known, however, a plasma centrifuge will not completely separate all of the particles in the aforementioned manner.
As indicated above in connection with Eq. 1, a force balance can be achieved for all conditions when the electric field E is chosen to confine ions, and ions exhibit confined orbits. In the plasma filter of the present invention, unlike a centrifuge, the electric field is chosen with the opposite sign to extract ions. The result is that ions of mass greater than a cut-off value, M.sub.c, are on unconfined orbits. The cut-off mass, M.sub.c, can be selected by adjusting the strength of the electric and magnetic fields. The basic features of the plasma filter can be described using the Hamiltonian formalism.
The total energy (potential plus kinetic) is a constant of the motion and is expressed by the Hamiltonian operator: EQU H=e.PHI.+(P.sub.r.sup.2 +P.sub.z.sup.2)/(2M)+(P.sub..theta. -e.PSI.).sup.2 /(2Mr2)
where P.sub.r =Mv.sub.r, P.sub..theta. =Mrv.sub..theta. +e.PSI., and P.sub.z =Mv.sub.z are the respective components of the momentum and e.PHI. is the potential energy. .PSI.=r.sup.2 B.sub.z /2 is related to the magnetic flux function and .PSI.=V.sub.ctr -.alpha..PSI. is the electric potential. E=-.gradient..PSI. is the electric field which is chosen to be greater than zero for the filter case of interest. We can rewrite the Hamiltonian: EQU H=eV.sub.ctr -e.alpha.r.sup.2 B.sub.z /2+(P.sub.r.sup.2 +P.sub.z.sup.2)/(2M)+(P.sub..theta. -er.sup.2 B.sub.z /2).sup.2 /(2Mr.sup.2).
We assume that the parameters are not changing along the z axis, so both P.sub.z and P.sub..theta. are constants of the motion. Expanding and regrouping to put all of the constant terms on the left hand side gives: EQU H-eV.sub.ctr -P.sub.z.sup.2 /(2M)+P.sub..theta..OMEGA./2=P.sub.r.sup.2 /(2M)+(P.sub..theta..sup.2 /(2Mr.sup.2)+(M.OMEGA.r.sup.2 /2)(.OMEGA./4-.alpha.)
where .OMEGA.=eB/M.
The last term is proportional to r.sup.2, so if .OMEGA./4-.alpha.&lt;0 then, since the second term decreases as 1/r.sup.2, P.sub.r.sup.2 must increase to keep the left-hand side constant as the particle moves out in radius. This leads to unconfined orbits for masses greater than the cut-off mass given by:
and where a is the radius of the chamber.
So, for example, normalizing to the proton mass, M.sub.p, we can rewrite Eq. 2 to give the voltage required to put higher masses on loss orbits: EQU V.sub.ctr &gt;1.2.times.10.sup.-1 (a(m)B(gauss)).sup.2 /(M.sub.C /M.sub.P).
Hence, a device radius of 1 m, a cutoff mass ratio of 100, and a magnetic field of 200 gauss require a voltage of 48 volts.
The same result for the cut-off mass can be obtained by looking at the simple force balance equation given by:
which differs from Eq. 1 only by the sign of the electric field and has the solutions: ##EQU2##
so if 4E/rB.sub.z.OMEGA.&gt;1 then .omega. has imaginary roots and the force balance cannot be achieved. For a filter device with a cylinder radius "a", a central voltage, V.sub.ctr, and zero voltage on the wall, the same expression for the cut-off mass is found to be: EQU M.sub.C =ea.sup.2 B.sup.2 /(8 V.sub.ctr).
Where B=B.sub.z in this case, and when the mass M of a charged particle is greater than the threshold value (M&gt;M.sub.c), the particle will continue to move radially outwardly until it strikes the wall, whereas the lighter mass particles will be contained and can be collected at the exit of the device. The higher mass particles can also be recovered from the walls using various approaches.
It is important to note that for a given device the value for M.sub.c in equation 3 is determined by the magnitude of the magnetic field, B, and the voltage at the center of the chamber (i.e. along the longitudinal axis), V.sub.ctr. These two variables are design considerations and can be controlled.
The discussion above has been specifically directed to the case where the magnetic field is oriented substantially parallel to the central longitudinal axis, and has only an axial component B.sub.z. For the case wherein the magnetic field has a helical configuration and, thus, has both an axial component, B.sub.z, and an azimuthal component, B.sub..theta., a similar analysis leads to slightly different result. The same derivation logic, however, still applies.
To evaluate the effect of the azimuthal component, B.sub..theta., of the magnetic field on the cut-off mass, M.sub.c, one can use the Hamiltonian formalism: EQU H=e.PHI.+P.sub.r.sup.2/ 2m+(P.sub.z -eA.sub.z).sup.2 /2m+(P.sub..theta. -e.PSI.).sup.2 /2mr.sup.2
where P.sub.r, P.sub.z, P.sub..theta., are respective components of canonical momentum, .PSI.=r.sup.2 B.sub.z /2 and A.sub.z =B.sub..theta. rln(r) are components of magnetic vector potential and .PHI.=V.sub.ctr -.alpha..PSI. is the electric potential. Taking into account that the azimuthal and axial components of momentum as well as total particle energy, H, are conserved, one can express the radial component of the momentum, P.sub.r, as a function of r: EQU 4 P.sub.r.sup.2 /(eBr.sub.o).sup.2 =(M/M.sub.c) (x-1)-(1-b.sup.2) (x-1).sup.2 /x-b.sup.2 ln.sup.2 x
where x=r.sup.2 /r.sub.o.sup.2, r.sub.o is initial coordinate of the particle, .OMEGA.=eB/m is ion cyclotron frequency, b=B.sub..theta. (r.sub.o)/B; B.sup.2 =B.sub..theta..sup.2 +B.sub.z.sup.2. As in the case of the standard filter the ion orbits can be unconfined (Pr monotonically increases with r) or confined (P.sub.r =0 at r&gt;r.sub.o) depending on the ratio M/M.sub.c which is defined by a mass of the ions. The additional term in the last equation somewhat increases the cut-off mass which can be described by the following approximate formula: EQU M.sub.c =(ea.sup.2 (B.sub.z.sup.2 +B.sub..theta..sup.2)/8V)(1-1.28b.sup.2 +1.49b.sup.3 -0.56b.sup.4) (Eq. 4)
which has accuracy better than 1% in the full range of b.sub.1 0&lt;b&lt;1. If the ratio B.sub..theta. /B&lt;1 then the cut-off mass is not very sensitive to the Bo and hence to the initial radial position of the ion in the filter. For example, if the source of the plasma is limited by the radii, 0.6a&lt;r&lt;a, and B.sub..theta. (a)/B.sub.z =0.3, one can expect variation of the cut-off mass of about 10% which is acceptable for separation of the ions with mass ratio of about 2.
It is also important to note that the addition of the azimuthal magnetic field component, B.sub..theta., creates a controllable axial plasma flow which has an axial velocity, v.sub.z, that can be expressed as: EQU v.sub.z=E.sub.r B.sub..theta. /B.sup.2.
At E.sub.r.about.r, the axial velocity has a flat radial profile. Further, the magnitude of the axial velocity, v.sub.z, is proportional to the axial current, I, that is flowing in the conductor or coil which generates the azimuthal component of the magnetic field, B.sub..theta.. It can be mathematically shown that this relationship is: EQU v.sub.z =10.sup.-7 eI/2M.sub.c.
Accordingly, the axial velocity, v.sub.z, of plasma flow through a filter can be controlled by variation of the current, I, that creates B.sub..theta..
In light of the above it is an object of the present invention to provide a plasma mass filter with a helical magnetic field which effectively separates low-mass charged particles from high-mass charged particles. It is another object of the present invention to provide a plasma mass filter with a helical magnetic field which has variable design parameters that permit the operator to select a demarcation between low-mass particles and high-mass particles. Still another object of the present invention is to provide a plasma mass filter with a helical magnetic field which allows the operator to control the axial velocity of the plasma through the filter. Yet another object of the present invention is to provide a plasma mass filter with a helical magnetic field which is easy to use, relatively simple to manufacture, and comparatively cost effective.